#### 5.3 Some notable features

Let us see how three singularities on the $u$-plane
move as we change $\mu $,
by solving (5.2.11). An example is shown in Fig. 5.10. On the right, the path in the
$\mu $
space is given. On the left, the three singularities for a given
$\mu $
is shown with three dots colored by red, green and blue connected to a triangle. As
$\mu $ moves along a semicircle
with constant, large $\left|\mu \right|$,
the quark point $u\sim {\mu}^{2}$
rotates the $u$-plane
once. At the same time, the monopole point and the dyon point of the eﬀective pure
$SU\left(2\right)$ theory rotates by
$90$ degrees, as we see from
(5.2.14). Now we make $\left|\mu \right|$
decrease ﬁrst; all three singularities come close to the origin of the
$u$-plane. Finally,
we make $\left|\mu \right|$
come back to the same semicircle again. As can be seen in the ﬁgure, this process exchanges the
quark point and the monopole point. We learned that, using the strongly-coupled region, we can
continuously change a quark into a monopole.

Finally, let us study the discriminant of the equation (5.2.11) itself, which is given by

$${\mu}^{3}+\frac{27}{8}{\Lambda}^{3}=0.$$ | (5.3.1) |

Take $\mu =-3\Lambda \u22152$
as an explicit choice. Then there is one singularity in the
$u$-plane at
$u=-15{\Lambda}^{2}\u22154$, and two singularities collide
at $u=3{\Lambda}^{2}$. In the curve, we ﬁnd
that the branch points of $x\left(z\right)$
consist of one at $z=\infty $ and
three colliding at $z=-1$.
See Fig. 5.11. From the curve, we immediately see that
$a={a}_{D}=0$,
since the integration cycles shrink. Using the BPS mass formula, we see that both
electrically charged particles and magnetically charged particles are simultaneously
becoming very light. This is a rather unusual situation for an eye trained in the classical
ﬁeld theory. Semiclassically, the magnetically charged particles come from solitons,
which are always parametrically heavier than the electrically charged particles which are
quanta of elementary ﬁelds in the theory. We will study this system in more details
Sec. 10.